To solve this problem, let's first convert each mixed number to an improper fraction, then perform the addition and subtraction. Finally, we’ll simplify the result.

Step 1: Convert Mixed Numbers to Improper Fractions

1. $3 \frac{2}{5}$: $3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$

2. $2 \frac{2}{3}$: $2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$

3. $5 \frac{7}{10}$: $5 \frac{7}{10} = \frac{5 \times 10 + 7}{10} = \frac{50 + 7}{10} = \frac{57}{10}$

Step 2: Calculate the Common Denominator and Perform the Operations

Now we have: $\frac{17}{5} + \frac{8}{3} + \frac{57}{10}$

The least common multiple of 5, 3, and 10 is 30. We’ll convert each fraction to have a denominator of 30:

1. $\frac{17}{5} = \frac{17 \times 6}{5 \times 6} = \frac{102}{30}$
2. $\frac{8}{3} = \frac{8 \times 10}{3 \times 10} = \frac{80}{30}$
3. $\frac{57}{10} = \frac{57 \times 3}{10 \times 3} = \frac{171}{30}$

Now, we can add these fractions: $\frac{102}{30} + \frac{80}{30} + \frac{171}{30} = \frac{353}{30}$

Step 3: Simplify the Result

The fraction $\frac{353}{30}$ is already in its simplest form since 353 and 30 have no common factors other than 1.

Final Answer

$\frac{353}{30}$

Would you like a more detailed breakdown of each step, or do you have any questions?

Here are some related questions for further exploration:

1. How can we convert improper fractions back to mixed numbers?
2. What are other methods to find the least common multiple?
3. How would we approach this problem if we wanted to work directly with mixed numbers?
4. Why is it important to simplify fractions to their lowest terms?
5. How can understanding mixed numbers help in real-life applications, like measuring ingredients?

Tip: Always convert mixed numbers to improper fractions when performing addition or subtraction—it makes the calculations straightforward.